To have yourself listed as the "Registrant", you need to have the whois records for the domain modified.
You do this by having whoever is the current Registrant or Administrator of the domain name edit those whois records to replace incorrect or outdated information with the "new" or "updated" (your) information.
You should start this process by contacting whoever registered the domain for you; they should know what to do. Things can get a bit more difficult if you no longer can contact that person.
Note that this only accomplishes making you the "Registrant" (which is the equivalent of "getting the domain in your name"), and has nothing to do with "transferring" where the domain is registered or hosted; that takes additional steps.
Usually a "transfer" between registrars involve a "renewal", where in conjunction with the transfer you pay for an additional year's (or multiple years') registration fee to the new registrar, and that time is added to whatever time is remaining on the current registration period. This process takes some time, so you don't want to late until the "last minute" to attempt a transfer for a domain that is about to expire.
Transferring hosting is simply a matter of arranging to have hosting set up (which includes adding the domain name to the new hosts nameservers) and modifying the domain's nameserver information on file with the registrar to point to the new host's nameservers.
There can also be circumstances where you need to move a domain between accounts at a given host, and this takes additional steps that vary with the circumstances and the host involved.
That's a "general overview" of the process; as lensman pointed out - we could give you better answers if we knw more about your situation and what you need to do.
What is the domain name in question? (from this we can locate the registrar and see where/how it is currently hosted, renewal status etc.)
Can you access the registrar's account management tools for that domain? ( from this we can see what you are able to modify yourself without requiring another's help or cooperation).